Question

Prove this identity using the product-to-sum identity for sin... sin^2 x=(1-cos(2x))/(2)

Answer 1

sin²x = (1 - cos2x)/2 ⇒ proved down

Explanation:

∵ sin²x = (sinx)(sinx) ⇒ add and subtract (cosx)(cosx)

(sinx)(sinx) + (cosx)(cosx) - (cosx)(cosx)

∵ (cosx)(cosx) - (sinx)(sinx) = cos(x + x) = cos2x

∴ - cos2x + cos²x = -cos2x + (1 - sin²x)

∴ 1 - cos2x - sin²x = (1 - cos2x)/2 ⇒ equality of the two sides

∴ (1 - cos2x) - 1/2(1 - cos2x) = sin²x

∴ 1/2(1 - cos2x) = sin²x

∴ sin²x = (1 - cos2x)/2